= Locally uniform convergence of holomorphic functions
= Weierstrass convergence theorem
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A <locally uniform convergence> limit of <holomorphic functions> is holomorphic, and its <derivatives> converge locally uniformly as well. The limit is continuous. On each small <triangle> contained in the <domain>, uniform convergence allows passage to the limit in its <contour integral>; the integrals vanish by the <Cauchy integral theorem>. The <Morera theorem> therefore makes the limit holomorphic. Applying the <Cauchy integral formula> on circles in a slightly larger compact subset proves locally uniform convergence of every fixed-order <derivative>.
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